Products of n open subsets in the space of continuous functions on [0,1]

Ehrhard Behrends. "Products of n open subsets in the space of continuous functions on [0,1]." Studia Mathematica 204.1 (2011): 73-95. <http://eudml.org/doc/285582>.

@article{EhrhardBehrends2011,
abstract = {Let O₁,...,Oₙ be open sets in C[0,1], the space of real-valued continuous functions on [0,1]. The product O₁ ⋯ Oₙ will in general not be open, and in order to understand when this can happen we study the following problem: given f₁,..., fₙ ∈ C[0,1], when is it true that f₁ ⋯ fₙ lies in the interior of $B_\{ε\}(f₁) ⋯ B_\{ε\}(fₙ)$ for all ε > 0 ? ($B_\{ε\}$ denotes the closed ball with radius ε and centre f.) The main result of this paper is a characterization in terms of the walk t ↦ γ(t): = (f₁(t),..., fₙ(t)) in ℝⁿ. It has to behave in a certain admissible way when approaching x ∈ ℝⁿ | x₁ ⋯ xₙ = 0. We will also show that in the case of complex-valued continuous functions on [0,1] products of open subsets are always open},
author = {Ehrhard Behrends},
journal = {Studia Mathematica},
keywords = {multiplication as a mapping; open mapping; Banach algebra},
language = {eng},
number = {1},
pages = {73-95},
title = {Products of n open subsets in the space of continuous functions on [0,1]},
url = {http://eudml.org/doc/285582},
volume = {204},
year = {2011},
}

TY - JOUR
AU - Ehrhard Behrends
TI - Products of n open subsets in the space of continuous functions on [0,1]
JO - Studia Mathematica
PY - 2011
VL - 204
IS - 1
SP - 73
EP - 95
AB - Let O₁,...,Oₙ be open sets in C[0,1], the space of real-valued continuous functions on [0,1]. The product O₁ ⋯ Oₙ will in general not be open, and in order to understand when this can happen we study the following problem: given f₁,..., fₙ ∈ C[0,1], when is it true that f₁ ⋯ fₙ lies in the interior of $B_{ε}(f₁) ⋯ B_{ε}(fₙ)$ for all ε > 0 ? ($B_{ε}$ denotes the closed ball with radius ε and centre f.) The main result of this paper is a characterization in terms of the walk t ↦ γ(t): = (f₁(t),..., fₙ(t)) in ℝⁿ. It has to behave in a certain admissible way when approaching x ∈ ℝⁿ | x₁ ⋯ xₙ = 0. We will also show that in the case of complex-valued continuous functions on [0,1] products of open subsets are always open
LA - eng
KW - multiplication as a mapping; open mapping; Banach algebra
UR - http://eudml.org/doc/285582
ER -